What is the value of cos(180)+isin(180) ?

The value of cos(180)+isin(180) is -1

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cos(180)+isin(180)

cos (18 0^{\circ}) + i sin (18 0^{\circ})

- 1

Solution steps

cos (18 0^{\circ}) + i sin (18 0^{\circ})

Use the following trivial identity:

cos (18 0^{\circ}) = (- 1)

cos (18 0^{\circ})

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= (- 1)

Use the following trivial identity:

sin (18 0^{\circ}) = 0

sin (18 0^{\circ})

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 0

= - 1 + i 0

Simplify

- 1 + i 0 : - 1

- 1 + i 0

Apply rule

0 \cdot a = 0

= - 1 + 0

Simplify

= - 1

= - 1

2 (1 + cos (1))

sin (8 0^{\circ}) cos (2 0^{\circ}) - cos (8 0^{\circ}) sin (2 0^{\circ})

cos^{2} (3 1^{\circ}) - sin^{2} (3 1^{\circ})

4 sin (2 (\frac{π}{6}))

arcsin (\frac{sin ( 6 0 ^{\circ} )}{1.5})